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By Irving Kaplansky

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T2)]m~\ with equality holding here if and only if the quadratic forms d2Hx and J 1H2 are proportional. Finally, let us note the following relationship between mixed curvature functions and mixed volumes [3] : Sm (Ti* T 2, . . , T2, E , . . , E), m-1 m S m (Ti, T i , . . , Tt1) H 2 d(ù = V (T2, Ti , . . , Ti, E , . . , E). In these formulas, E denotes a unit ball. §2. GENERALIZED SOLUTION OF THE MINKOWSKI PROBLEM In this section, we present the classical result of Minkowski on the problem of existence and uniqueness of a convex hypersurface with preassigned Gaussian curvature K (v).

I. Statement of the problem. Uniqueness of the solution. Let K (v) denote a positive continuous function defined on the hypersphere £2. Minkowski’s problem is the problem of existence of a convex hypersurface F with Gaussian curvature K (v) at a point with exterior normal v. The Gaussian curvature is under­ stood in the sense of the definition given in Sec. I, subsection 2. Specifically, for a given hypersurface F and a point x E F9 let us consider the ratio (o (G)IS (G), where S(G) is the area o f a domain G containing x and co(G) is the spherical image of G.

We have к к m Bk where e is the unit vector in the direction The function K(X) is continuous, so that there exists an a such that £(£) < a. Hence, « “ >■¿■ 2 1 1 (<*)*>Iк m Sk Denote by M m the set of domains g™ in which I %e\ > e > 0. Then Sm> When e is sufficiently small and m sufficiently large, the diameters of the domains g™are small and Mm GENERALIZED SOLUTION OF THE MINKOWSKI PROBLEM 31 where S is the area o f the hypersphere SI. Thus, when m is sufficiently large, the area of the projection of the polyhedron Pm onto the hyperplane a is ^ 4a Perform a symmetrization of the polyhedron P m with respect to the hyperplane a.

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An Introduction to Differential Algebra by Irving Kaplansky


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